Micky van de Ven emerged as the first backline player to score two goals in a English top-flight match for Tottenham since the Belgian defender in March 2013.
At a glance
The Spurs defender nods into the net twice in opening period
Everton's Jake O'Brien has goal disallowed after VAR review
Pape Matar Sarr scores third in dying moments
The defeat is Everton's initial at Hill Dickinson Stadium
The Tottenham defender found the net twice as Tottenham prolonged their unbeaten run on the road in the Premier League and handed Everton their first defeat at their new home.
The Toffees were unbeaten in their last five matches at their new ground at Bramley-Moore Dock, having left their historic home behind after more than a century at the end of last season.
But Spurs captain Van de Ven struck twice from corners in the opening 45 minutes to put their strong form at risk, and they were unable to mount a comeback before Pape Matar Sarr - set up by ex-Everton forward Richarlison - headed home a third to seal the victory.
Jake O'Brien believed he had equalized the tally a short while after the defender's opener, glancing a bullet header past the Spurs keeper from an set-piece.
The Portuguese striker, meanwhile, failed to convert a handful of chances, while the Italian goalkeeper made a series of fine saves to guarantee the London club went five matches on the road without defeat in the English top flight.
But the video assistant referee stepped in, advising Craig Pawson to review the goal on the screen and the referee ruled that the winger and Iliman Ndiaye were in an illegal position and had impeded with the Spurs keeper.
The win moves Thomas Frank's side into third place in the Premier League, five points off north London rivals and leaders Arsenal.
The Merseyside club stay in 14th position and are half a dozen points ahead of the relegation zone.
After a drab midweek performance in the Champions League against Monaco, where they mustered a mere two shots on target, Frank opted for the French striker up front instead of ex-Everton forward the Brazilian.
The summer signing was failed to make much of an impact before being substituted for Richarlison after 60 minutes on the field, but it turns out he proved unnecessary in the end.
Defender the Dutchman transformed into an attacking force to be feared.
Spurs' first goal came from a well-worked set-piece play, with Rodrigo Bentancur heading the ball toward the six-yard box for an free Van de Ven to score.
And the captain - stepping up as captain in Cristian Romero's unavailability - notched his next goal in added time as he outjumped Jordan Pickford to head home.
The Dutchman was just as outstanding in defence, combining effectively with Kevin Danso as the pair dealt superbly with the 34 crosses sent their way by Everton.
Everton's recent stadium had been a happy hunting ground, but it seems the novelty may have faded somewhat and supporters flocked to the gates after Sarr's third for Tottenham.
The end in Everton's winning streak at home will not be a massive worry for David Moyes, but their striker situation will be.
Beto and Thierno Barry - the Toffees' pair of recognised centre-forwards - had netted just a single goal combined before Sunday's outing.
And they were unable to add to that poor record on a wet and gloomy evening on in Liverpool.
The striker's form was summed up when he chose to bicycle-kick a golden opportunity from near the goal, even with having room and time to readjust for a simpler shot that would have challenged the keeper.
The substitute, as well, failed to influence the game after replacing the starter in the second half.
It was not for a shortage of chance. Most of Everton's offensive moves came through Grealish and the forward, who relentlessly whipped dangerous balls into the Tottenham penalty area.
Everton: The team travel to the Stadium of Light in the Premier League on Monday, 3 November at 20:00 GMT.
Spurs: The London club travel to Newcastle's ground on Wednesday, 29 October for their fourth-round Carabao Cup tie against the Magpies.